A247612 a(n) = Sum_{k=0..7} binomial(14,k)*binomial(n,k).
1, 15, 120, 680, 3060, 11628, 38760, 116280, 316767, 788161, 1805100, 3840420, 7660250, 14446134, 25947612, 44668692, 74091645, 118941555, 185495056, 281936688, 418766304, 609260960, 869994720, 1221419808, 1688512539, 2301487461, 3096583140, 4116923020
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
- C. Krattenthaler, Advanced determinant calculus Séminaire Lotharingien de Combinatoire, B42q (1999), 67 pp, (see p. 54).
- Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).
Programs
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Magma
[(1680+386012*n-958048*n^2+943761*n^3-455455*n^4+123123*n^5- 17017*n^6+1144*n^7)/1680: n in [0..40]];
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Magma
I:=[1,15,120,680,3060,11628,38760, 116280]; [n le 8 select I[n] else 8*Self(n-1)-28*Self(n-2)+56*Self(n-3)-70*Self(n-4)+56*Self(n-5) -28*Self(n-6)+8*Self(n-7)-Self(n-8): n in [1..40]];
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Mathematica
Table[(1680 + 386012 n - 958048 n^2 + 943761 n^3 - 455455 n^4 + 123123 n^5 - 17017 n^6 + 1144 n^7)/1680, {n, 0, 40}] (* or *) CoefficientList[Series[(1 + 7 x + 28 x^2 + 84 x^3 + 210 x^4 + 462 x^5 + 924 x^6 + 1716 x^7)/(1 - x)^8, {x, 0, 40}], x] LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{1,15,120,680,3060,11628,38760,116280},30] (* Harvey P. Dale, May 12 2017 *) -
Sage
m=7; [sum((binomial(2*m,k)*binomial(n,k)) for k in (0..m)) for n in (0..40)] # Bruno Berselli, Sep 22 2014
Formula
G.f.: (1 + 7*x + 28*x^2 + 84*x^3 + 210*x^4 + 462*x^5 + 924*x^6 + 1716*x^7) / (1-x)^8.
a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8).
a(n) = (1680 + 386012*n - 958048*n^2 + 943761*n^3 - 455455*n^4 + 123123*n^5 - 17017*n^6 + 1144*n^7)/1680.
Extensions
Definition edited by Robert Israel, Sep 22 2014